Community Detection via Maximization of Modularity and Its Variants

In this paper, we first discuss the definition of modularity (Q) used as a metric for community quality and then we review the modularity maximization approaches which were used for community detection in the last decade. Then, we discuss two opposite yet coexisting problems of modularity optimization: in some cases, it tends to favor small communities over large ones while in others, large communities over small ones (so called the resolution limit problem). Next, we overview several community quality metrics proposed to solve the resolution limit problem and discuss Modularity Density (Qds) which simultaneously avoids the two problems of modularity. Finally, we introduce two novel fine-tuned community detection algorithms that iteratively attempt to improve the community quality measurements by splitting and merging the given network community structure. The first of them, referred to as Fine-tuned Q, is based on modularity (Q) while the second one is based on Modularity Density (Qds) and denoted as Fine-tuned Qds. Then, we compare the greedy algorithm of modularity maximization (denoted as Greedy Q), Fine-tuned Q, and Fine-tuned Qds on four real networks, and also on the classical clique network and the LFR benchmark networks, each of which is instantiated by a wide range of parameters. The results indicate that Fine-tuned Qds is the most effective among the three algorithms discussed. Moreover, we show that Fine-tuned Qds can be applied to the communities detected by other algorithms to significantly improve their results

A smart local moving algorithm for large-scale modularity-based community detection

We introduce a new algorithm for modularity-based community detection in large networks. The algorithm, which we refer to as a smart local moving algorithm, takes advantage of a well-known local moving heuristic that is also used by other algorithms. Compared with these other algorithms, our proposed algorithm uses the local moving heuristic in a more sophisticated way. Based on an analysis of a diverse set of networks, we show that our smart local moving algorithm identifies community structures with higher modularity values than other algorithms for large-scale modularity optimization, among which the popular 'Louvain algorithm' introduced by Blondel et al. (2008). The computational efficiency of our algorithm makes it possible to perform community detection in networks with tens of millions of nodes and hundreds of millions of edges. Our smart local moving algorithm also performs well in small and medium-sized networks. In short computing times, it identifies community structures with modularity values equally high as, or almost as high as, the highest values reported in the literature, and sometimes even higher than the highest values found in the literature

On Variants of CM-triviality

We introduce a generalization of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalizes one-basedness. We show that, under this condition, a stable field is internal to the family, and a group of finite Lascar rank has a normal nilpotent subgroup such that the quotient is almost internal to the family

The imperfect hiding : some introductory concepts and preliminary issues on modularity

In this work we present a critical assessment of some problems and open questions on the debated notion of modularity. Modularity is greatly in fashion nowadays, being often proposed as the new approach to complex artefact production that enables to combine fast innovation pace, enhanced product variety and reduced need for co-ordination. In line with recent critical assessments of the managerial literature on modularity, we sustain that modularity is only one among several arrangements to cope with the complexity inherent in most high-technology artefact production, and by no means the best one. We first discuss relations between modularity and the broader (and much older within economics) notion of division of labour. Then we sustain that a modular approach to labour division aimed at eliminating technological interdependencies between components or phases of a complex production process may have, as a by-product, the creation of other types of interdependencies which may subsequently result in inefficiencies of various types. Hence, the choice of a modular design strategy implies the resolution of various tradeoffs. Depending on how such tradeoffs are solved, different organisational arrangements may be created to cope with ‘residual’ interdependencies. Hence, there is no need to postulate a perfect isomorphism, as some recent literature has proposed, between modularity at the product level and modularity at the organisational level

Multistep greedy algorithm identifies community structure in real-world and computer-generated networks

We have recently introduced a multistep extension of the greedy algorithm for modularity optimization. The extension is based on the idea that merging l pairs of communities (l>1) at each iteration prevents premature condensation into few large communities. Here, an empirical formula is presented for the choice of the step width l that generates partitions with (close to) optimal modularity for 17 real-world and 1100 computer-generated networks. Furthermore, an in-depth analysis of the communities of two real-world networks (the metabolic network of the bacterium E. coli and the graph of coappearing words in the titles of papers coauthored by Martin Karplus) provides evidence that the partition obtained by the multistep greedy algorithm is superior to the one generated by the original greedy algorithm not only with respect to modularity but also according to objective criteria. In other words, the multistep extension of the greedy algorithm reduces the danger of getting trapped in local optima of modularity and generates more reasonable partitions.Comment: 17 pages, 2 figure

The imperfect hiding: Some introductory concepts and preliminary issues on modularity.

In this work we present a critical assessment of some problems and open questions on the debated notion of modularity. Modularity is greatly in fashion nowadays, being often proposed as the new approach to complex artefact production that enables to combine fast innovation pace, enhanced product variety and reduced need for co-ordination. In line with recent critical assessments of the managerial literature on modularity, we sustain that modularity is only one among several arrangements to cope with the complexity inherent in most high-technology artefact production, and by no means the best one. We first discuss relations between modularity and the broader (and much older within economics) notion of division of labour. Then we sustain that a modular approach to labour division aimed at eliminating technological interdependencies between components or phases of a complex production process may have, as a by-product, the creation of other types of interdependencies which may subsequently result in inefficiencies of various types. Hence, the choice of a modular design strategy implies the resolution of various tradeoffs. Depending on how such tradeoffs are solved, different organisational arrangements may be created to cope with 'residual' interdependencies. Hence, there is no need to postulate a perfect isomorphism, as some recent literature has proposed, between modularity at the product level and modularity at the organisational level.

Spectral Graph Forge: Graph Generation Targeting Modularity

Community structure is an important property that captures inhomogeneities common in large networks, and modularity is one of the most widely used metrics for such community structure. In this paper, we introduce a principled methodology, the Spectral Graph Forge, for generating random graphs that preserves community structure from a real network of interest, in terms of modularity. Our approach leverages the fact that the spectral structure of matrix representations of a graph encodes global information about community structure. The Spectral Graph Forge uses a low-rank approximation of the modularity matrix to generate synthetic graphs that match a target modularity within user-selectable degree of accuracy, while allowing other aspects of structure to vary. We show that the Spectral Graph Forge outperforms state-of-the-art techniques in terms of accuracy in targeting the modularity and randomness of the realizations, while also preserving other local structural properties and node attributes. We discuss extensions of the Spectral Graph Forge to target other properties beyond modularity, and its applications to anonymization